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Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
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01 Jan 2004
TL;DR: For a generic pair of polynomials P of degree n and Q of degree m, where m, n are satisfying some conditions, P(f)=Q(g) for meromorphic functions f,g implies f=const, g=const as discussed by the authors.
Abstract: We prove that for a generic pair (P, Q) of polynomials P of degree n and Q of degree m, where m, n are satisfying some conditions, P(f)=Q(g) for meromorphic functions f,g implies f=const, g=const. We also give another proof of the statement saying that a generic polynomial of degree at least 5 is a uniqueness polynomial for meromorphic functions.
22 citations
TL;DR: The Galois group of the generalized Laguerre polynomial F 2 n = e x x −2 n ( d 2 n ( e − x x 4 n )/ dx 2 n ) is the alternating group of degree 2 n, provided that F 2n is irreducible over the rationals as discussed by the authors.
22 citations
TL;DR: For any cyclic group G of odd order and any field F whose characteristic does not divide the order of G, it is possible to find a polynomial which parametrizes all extensions of F with group G as discussed by the authors.
Abstract: For any cyclic group G of odd order and any field F whose characteristic does not divide the order of G it is possible to find a polynomial which parametrizes all extensions of F with group G. In this paper such a polynomial is explicitly constructed for each such group G which is valid for all such fields F.
22 citations
TL;DR: In this paper, it was shown that the action of the absolute Galois group on dessins d'enfants of given genus g is faithful, a result that had been previously established for g = 0 and g = 1.
Abstract: We show that the action of the absolute Galois group on dessins d’enfants of given genus g is faithful, a result that had been previously established for g = 0 and g = 1.
22 citations
TL;DR: In this article, a general formula for the degree of the reflex field is given, and complete lists of p-structures and reflex fields are provided for (K: Cl ) = 2 n, with n = 3, 4, 5 and 7.
Abstract: A CM-field K defines a triple (C, H, p), where C is the Galois group of the Galois closure of K, H is the subgroup of C fixing K, and p E C is induced by complex conjugation. A '4p-structure" identifies CM-fields when their triples are identified under the action of the group of automorphisms of C. A classification of the p-structures is given, and a general formula for the degree of the reflex field is obtained. Complete lists of p-structues and reflex fields are provided for ( K: Cl ) = 2 n, with n = 3, 4, 5 and 7. In addition, simple degenerate Abelian varieties of CM-type are constructed in every composite dimension. The collection of reflex fields is also determined for the dihedral group C = D2n, with n odd and H of order 2, and a relative class number formula is found.
22 citations